Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P15 doc

Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality ApproachKazuo Tanaka, Hua O.. Moreover, the problem of H control of this⬁ class of nonlinear time-delay systems is c

Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach

Kazuo Tanaka, Hua O Wang Copyright 䊚 2001 John Wiley & Sons, Inc.

ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic

CHAPTER 15

FUZZY CONTROL OF NONLINEAR

TIME-DELAY SYSTEMS

In this chapter, a class of nonlinear time-delay systems based on the

Sugeno T-S fuzzy model is defined 1 We investigate the delay-indepen-dent stability of this model A model-based fuzzy stabilization design utilizing

the concept of parallel distributed compensation PDC is employed The main idea of the controller design is to derive each control rule to compen-sate each rule of a fuzzy system Moreover, the problem of H control of this⬁

class of nonlinear time-delay systems is considered The associated control

synthesis problems are formulated as linear matrix inequality LMI prob-lems

In the original T-S fuzzy model formulation, there is no delay in the control and state However, time delays often occur in many dynamical systems such as biological systems, chemical systems, metallurgical processing systems, and network systems Their existence is frequently a cause of instability and poor performance The study of stability and stabilization for

linear time-delay systems has received considerable attention 2᎐6 But these efforts were mainly restricted to linear time-delay systems Thus, it is impor-tant to extend the stability and stabilization issues to nonlinear time-delay systems In this chapter, a particular class of nonlinear time-delay systems is introduced based on the Tagaki-Sugeno fuzzy model This kind of nonlinear system is represented by a set of linear time-delay systems We will call this a

T-S model with time delays T-SMTD In the literature, the problem of stability and stabilization of time-delay systems has been dealt with a number

of different ways There are some results that are independent of the size of

w x the time delays in 2᎐4 , and the stability is satisfied for any value of the time delays There are also some delay-dependent results, in which the stability is

291

w x guaranteed up to some maximum value for the time delays 5, 6 This chapter is concerned with the problems of delay-independent stability and stabilization of T-S fuzzy models with time delays Particularly, we will employ the concept of parallel distributed compensation to study these problems Several new results concerned with the stability and stabilization of T-SMTD are derived Also, a sufficient condition for the H control of this⬁

model is given All the synthesis problems are formulated as LMIs, thus they are numerically efficient

Throughout the chapter, the notation M ) 0 will mean that M is a

positive definite symmetric matrix The symbol p will be used for premise

variables as in Chapters 12 and 13

15.1 T-S FUZZY MODEL WITH DELAYS AND STABILITY

CONDITIONS

15.1.1 T-S Fuzzy Model with Delays

To begin with, we represent a given nonlinear plant by the Takagi-Sugeno fuzzy model Then, we will define a new kind of model, the Takagi-Sugeno fuzzy model with time delays The main feature of the T-S fuzzy model is to

express the joint dynamics of each fuzzy implication rule by a linear system model Specifically, the Takagi-Sugeno fuzzy system is described by fuzzy IF-THEN rules, which locally represent linear input-output relations of a system The fuzzy system is of the following form:

Dynamic Part: Rule i

IF p t is M , , and p t is M ,1 i1 l i l

THEN

x t s A x t q B u t ,Ž Ž Ž i s 1, 2, , r Ž15.1.

Output Part: Rule i

IF p t is M , , and p t is M ,1 i1 l i l

THEN

y t s C x t Ž i Ž

Ž Ž Ž Ž

Here, x t , u t , y t , and p t respectively denote the state, input, output,

and parameter vectors The jth component of p t is denoted by p t , and

T-S FUZZY MODEL WITH DELAYS AND STABILITY CONDITIONS 293

the fuzzy membership function associated with the ith rule and jth

parame-Ž ter component is denoted by M Each p t is a measurable time-varying i j j

quantity In general, these parameters may be functions of the state variables, external disturbances, andror time

Ž There are two functions of p t associated with each rule The first function is called the truth value The truth value for the ith rule is defined

by the equation

l

␻ p t s iŽ Ž ŁM i jŽp t jŽ

js1

Throughout this chapter, we will assume that each ␻ is a nonnegativei

function and that the truth value of at least one rule is always nonzero The second function is called the firing probability The firing probability for the

ith rule is defined by the equation

␻ p t iŽ Ž

h iŽ p tŽ s r ,

Ýis1 ␻ p t iŽ Ž

where r denotes the number of rules in the rule base Under the previously

stated assumptions, this is always a well-defined function taking values between 0 and 1, and the sum of all the firing probabilities is identically equal to 1

Now, we introduce time delays into the above T-S fuzzy model Here, we assume there are time delays in both the state and control of the dynamic part Then, the i rule of the dynamic part of T-S fuzzy model becomes:

Rule i

IF p t is M , , and p t is M1 i1 l i l

THEN

x t s A x t q A x t yŽ Ž Ž ␶ q B u t q B u t y ␶ , Ž Ž

i s 1, 2, , r , Ž15.2.

where 0 F␶ - ⬁ and 0 F ␶ - ⬁ are the size of the time delays The initial1 2

Ž

condition is x t s 0, where t- 0

We call this model the T-S model with time delays T-SMTD In the following we will investigate the stability and design issues, such as delay-independent stabilization and H control, of this system.⬁

The dynamics described by the T-SMTD evolve according to the system of equations

r

x t sŽ h Žp A x t q A x t yŽ Ž ␶ .

is1

qB u t q B u t y i0 Ž i d Ž ␶2 4, Ž15.3.

r

y t sŽ Ýh iŽp C x t . i Ž

is1

The open-loop system is of the form

r

x t sŽ h Žp A x t q A x t yŽ Ž ␶ 4 Ž15.4.

is1

Remark 45 Our proposed model description can also be viewed as parame-ter-dependent interpolation between linear models; however, the exact classi-fication of the resultant system depends on the nature of the parameters For example, if each p is a known function of time, then the T-S model describes i

a linear time-varying system If, on the other hand, each p is a function of i

the state variables, then the T-S model describes an autonomous nonlinear system

15.1.2 Stability Analysis via Lyapunov Approach

A sufficient delay-independent stability condition for the open-loop system

Ž15.4 is given as follows:

THEOREM 58 The open-loop T-S fuzzy system with time delays 15.4 is globally asymptotically stable if there exist two common positi®e definite matrices

P and R such that

PA q A i0 T i0 P q PA R i d y1A T i d P q R - 0, i s 1, 2, , r, Ž15.5.

that is, two common matrices P and R ha®e to exist for all subsystems.

Proof. For the open-loop system 15.4 , we define a Lyapunov function as the following:

t

V x s x tŽ Ž Px t qŽ H x sŽ Rx s ds.Ž Ž15.6.

ty␶

T-S FUZZY MODEL WITH DELAYS AND STABILITY CONDITIONS 295

The derivate of V x along the open-loop system 15.4 is

r

˙

V x sŽ Ýh iŽp x t Ž PA q A P x t i0 i0 Ž

is1 r

T

q2Ýh iŽp x t Ž PA x t y i d Ž ␶1.

is1

qx tŽ Rx t y x t yŽ Ž ␶1. Rx t yŽ ␶ 1 Ž15.7. Using the fact that

2x tŽ PA x t y i d Ž ␶ F x t PA R A Px t1 Ž i d i d Ž

T

qx t yŽ ␶1. Rx t yŽ ␶ ,1 Ž15.8.

we have

r

T

˙

V x FŽ Ýh iŽp x t Ž

is1

= PA q A T

P q PA Ry 1A T P q R x tŽ -0, ᭙ x/0 15.9Ž

ŽQ.E.D

Remark 46 The system 15.4 is also said to be quadratically stable and the

Ž

function V x is called a quadratic Lyapunov function Theorem 58 thus

presents a sufficient condition for quadratic stability of the open-loop system

Ž15.4

15.1.3 Parallel Distributed Compensation Control

w x

In 7 , Wang et al utilized the concept of parallel distributed compensation

ŽPDC to design fuzzy controllers to stabilize fuzzy system 15.1 The idea is Ž

to design a compensator for each rule of the fuzzy model The resulting overall fuzzy controller, which is nonlinear in general, is a fuzzy blending of each individual linear controller The fuzzy controller shares the same fuzzy

sets with the fuzzy system 15.1 Here, we will apply the same controller structure to the T-SMTD, so the ith control rule is as follows:

Control Rule i

IF p t is M and, , and p t is M ,1 i1 l i l

THEN u t s yF x t , i s 1, , r.

The output of the PDC controller is determined by the summation

r

u t s yŽ Ýh iŽp F x t . i Ž Ž15.10.

is1

Note that the controller 15.10 is nonlinear in general

The Closed-Loop System Substituting 15.10 into 15.3 , we obtain the corresponding closed-loop system

r

2

x t s h p G x t q A x t y ␶ y B F x t y ␶

is1

r G q G i j ji A x t y i d Ž ␶ q A x t y ␶1. jd Ž 1.

q2Ý Ýh p h i j p ½ 2 x t q 2

is1 i -j

yB F x t y i d j ␶ y B F x t y ␶2 jd i 2

15.11

where G s A y B F i j i0 i0 j

15.2 STABILITY OF THE CLOSED-LOOP SYSTEMS

Now, we present a delay-independent stability condition for the closed-loop

system 15.11

THEOREM 59 If there exist matrices P ) 0, R ) 0, and R ) 0 such that1 2

the following matrix inequalities are satisfied, the closed-loop system 15.11 is quadratically stable:

PG q G i i i i T P q PA R i d y1 1 A T i d P q R q PB F P1 i d i y1Ry1 2 Py1F i T B i d T P q PR P2 - 0,

i s 1, , r , Ž15.12.

T

G q G i j ji G q G i j ji 1

Pž 2 / žq 2 / P q 2P A RŽ i d 1 A q A R i d jd 1 A jd.P

1

y 1 y 1 y 1 T T

qR q1 P B F PŽ i d j R2 P F B j i d

2

qB F Py 1Ry 1Py 1F T B T.qPR P F 0. Ž15.13.

STATE FEEDBACK STABILIZATION DESIGN VIA LMIs 297

Proof. Define the following Lyapunov function for the closed-loop system:

t

V x s x tŽ Ž Px t qŽ H x sŽ R x s ds1 Ž

ty␶ 1

qH x sŽ PR Px s ds.2 Ž Ž15.14.

ty␶ 2

Ž Taking the derivative of V x along the closed-loop system and using the

fact that for any vector x and x and matrix Y1 2

x1T Yx q x2 T2Y T x F x1 T1YRy 1Y T x q x1 T2Rx ,2 Ž15.15. where R is a positive definite matrix, we have

r

T

˙

V x sŽ Ýh iŽp x t Ž PG q G P q PA R i i i i i d 1 A P q R i d 1

is1

qPB F P i d i y 1Ry 2 1Py 1F i T B i d T P q PR P x t2 4 Ž

T

r T G q G i j ji G q G i j ji

q2is1 iÝ Ý-j h h x t i j Ž ½Pž 2 / žq 2 / P

1

q P A RŽ i d 1 A q A R i d jd 1 A jd.P q R1

2 1

y 1 y 1 y 1 T T

q ŽB F P i d j R2 P F B j i d

2

qB F P jd i y 1Ry 2 1Py 1F i T B T jd.qPR P x t 2 5 Ž

15.16

Since Ýr is1 h i ) 0 and h G 0, we have i

˙

ŽQ.E.D

15.3 STATE FEEDBACK STABILIZATION DESIGN VIA LMIs

The state feedback stabilization design problem can be stated as follows: Given a plant described by a T-SMTD model, find a PDC control that quadratically stabilizes the closed-loop system The design variables in this

problem are the gain matrices F 1 F i F r The following theorem states

conditions that are sufficient for the existence of such a PDC controller Taken together, these conditions form an LMI feasibility problem If this problem is analyzed numerically and a feasible solution is found, then a set of stabilizing gain matrices can be computed directly from the solution data

THEOREM 60 A sufficient condition for the existence of a PDC controller

that quadratically stabilizes the T-SMTD model 15.3 is that there exist matrices

X ) 0, W ) 0, W ) 0, and M , 1 F i F r, such that the following two LMI1 2 i conditions hold:

Ž a For e®ery 1 F i F r, the following equation is satisfied:

A X q XA q A W A i0 i0 i d 1 i d

X B M i d i

T T

ž yB M y M B q W i0 i i i0 2 /

- 0 15.18Ž

T T

M B i i d 0 yW2

Ž b For e®ery pair of indices satisfying 1 F i F j F r, the equation

U q V q W i j i j i j X B M i d j B M jd i

1

F0 Ž15.19.

T T

M B j i d 0 yW2 0

T T

holds, where

U s A X q XA i j i0 T i0qA X q XA j0 T j0,

V s yB M y M i j i0 j j T B y B M y M i0 j0 i i T B , j0

W s A W A i j i d 1 T i dqA W A jd 1 T jdq2W 2

Furthermore, if the matrices exist which satisfy these inequalities, then the feedback gains F s M X i i y1 will pro®ide a quadratically stabilizing PDC controller.

Proof. Let P s Xy 1, W s R1 y11, and W s R Then we can get the above2 2

H CONTROL⬁ 299

In this section, we will investigate the problem of disturbance rejection for the T-S fuzzy model with time delays We assume theith rule of the model is

IF p t is M and, , and p t is M ,1 i1 l i l

THEN

x t s A x t q A x t yŽ Ž Ž ␶ q B u t q B u t y ␶ q D w t , Ž Ž Ž

i s 1, 2, , r , Ž15.20.

z t s E x t ,Ž i Ž

where w t is the square integrable disturbance input vector and z t is the

controlled output

Our objective here is to construct an H controller in the form 15.10 such⬁

Ž

that a the controller is a stabilizer for the nonlinear time-delay system and

Ž b subject to assumption of zero initial condition, the controlled output z

⬁5 Ž 52 2w ⬁5 Ž 52 x w x

satisfies H0 z t F␥ H w t dt for all w g L 0 ⬁ , where ␥ is a pre-0 2

specified positive constant If this kind of controller exists, the nonlinear

time-delay system 15.20 is said to be stabilizable with an H -norm bound⬁ ␥

THEOREM 61 For the system 15.20 , a sufficient condition for the existence

of a PDC controller that stabilizes the T-SMTD model with an H -norm bound⬁ ␥

is that there exist matrices X ) 0, W ) 0, W ) 0, and M , 1 F i F r, such that1 2 i the following two LMI conditions hold:

Ž 1 For e®ery 1 F i F r, the equation

T

H i i X B M i d i D i XE i

T T

T

where

H s A X q XA i i i0 T i0qA W A i d 1 T i dyB M y M i0 i i T B i0 TqW ,2

is satisfied.

Ž 2 For e®ery pair of indices satisfying 1 F i F j F r, the equation

T

U q V q W i j i j i j X B M i d j B M jd i D i j XE i j

1

T T

F0 Ž15.22.

T T

T

holds, where

U s A X q XA i j i0 T i0qA X q XA j0 T j0,

V s yB M y M T B TyB M y M T B T

,

W s A W A i j i d 1 T i dqA W A jd 1 T jdq2W ,2

1

D s D D q D D i j i j j i ,

1

E s E E q E E i j i j j i

Furthermore, if matrices exist which satisfy these inequalities, then the feedback gains are gi®en by F s M X i i y 1

15.5 DESIGN EXAMPLE

Consider the following simple T-S fuzzy model with time delays where the fuzzy rules are given by

Rule 1

IF x t is M2 1 e.g., Small

THEN

x t s A x t q A x t yŽ Ž Ž ␶ q B u t q B u t y ␶ Ž Ž

Rule 2

IF x t is M2 2 e.g., Big

THEN

x t s A x t q AŽ Ž x t yŽ ␶ q B u t q B u t y ␶ Ž Ž

˙

DESIGN EXAMPLE 301

Ž w Ž Ž xT

Here x t s x t1 x t2 and

A1 sA2 s , B s B s10 20 ,

0.2

B s B1 2 s

0 This system is unstable for some initial conditions as shown in Figure 15.1 for

Ž w xT

the initial condition x t s 2 2 Now we want to design a PDC controller

to stabilize this system Using Theorem 60, we obtain the feedback gains of the PDC controller:

F s 11.221 12.87 ,

F s 8.872 12.33

Ž w xT The closed-loop response for the initial condition x t s 2 2 is shown in Figure 15.2 In the simulations,␶ and ␶ are chosen to 1 though they can be1 2

of different values

Fig 15.1 Response of the open-loop system.

Fig 15.2 Response of the closed-loop system.

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